# Mathematical Optimization Solutions

Advanced analytical techniques
What is Mathematical Optimization?

Mathematical Optimization, also known as Mathematical Programming, is an aid for decision making utilized on a grand scale across all industries. Advanced analytical techniques are used to find the best value of the inputs from a given set which is specified by physical limits of the problem and user's restrictions. The quality of the result is measured by a user metric provided as a scalar function of the inputs. Optimization problems come from a massively diverse range of fields and industries, such as portfolio optimization or calibration in finance, structural optimization in engineering, data fitting in weather forecasting, parameter estimation in chemistry and many more.

Optimization and NAG - Reap the Benefits

Whether the optimization problem is fitting data obtained from a particle accelerator or rebalancing your investment portfolio the solvers used to deliver the results need to be robust, reliable and thoroughly tested. NAG optimization experts have developed and extensively tested a wide range of routines that provide quick and accurate solutions to optimization problems. NAG optimization solvers are highly flexible, callable from many programming languages, environments and mathematical packages, and fully documented to simplify their deployment in your application. By embedding NAG software, analysts and software engineers are able to spend more time in other areas of their work, improving productivity and time management.

Fast and accurate solutions to optimization problems

There might be more than one way to formulate an optimization problem into a mathematical model and each type of model requires a specific optimization solver. Our aim is to offer you a comprehensive collection of optimization solvers so that you don't need to look anywhere else. NAG solvers are backed by nearly five decades of experience in developing numerical software and are supported by collaborations with many leading academics and universities. They cover a wide set of problems and circumstances so users do not feel limited by their model.

The main classes of optimization problems covered in the NAG Library are:

• Linear Programming (LP) – dense and sparse;
• Quadratic Programming (QP) – convex and nonconvex, dense and sparse;
• Nonlinear Programming (NLP) – dense and sparse, based on active-set SQP methods and interior point method (IPM);
• Global Nonlinear Programming – algorithms based on branching, multistart and stochastic optimization;
• Mixed Integer Nonlinear Programming (MINLP) – for dense (possibly nonconvex) problems;
• Semidefinite Programming (SDP) – both linear matrix inequalities (LMI) and bilinear matrix inequalities (BMI);
• Derivative-free Optimization (DFO) – solvers for problems where derivatives cannot be easily computed and finite difference approximation is not suitable;
• Least Squares (LSQ), data fitting, calibration, regression – linear and nonlinear, constrained and unconstrained.

For a full overview of the offered functionality in the NAG Library, please see the chapter introductions of the following Chapters in the Library:

In the NAG Library documentation there is further classification of the optimization problems and additional details are discussed to assist choosing the right solver for your specific requirements, in particular, data sparsity, smoothness and differentiability, and key features of various methods.

Optimization Consultancy

Sometimes an “out of the box” solution does not fully exploit the potential of your real-world problem. For these situations NAG offers direct access to our optimization team in the form of the Mathematical Optimization Consultancy service so organizations gain from our knowledge and experience. The problem will be discussed, evaluated, analysed and solutions offered to correctly advance your application. If you have a problem that could benefit from expert analysis and advice, contact us.

Benefits
Speed & Accuracy Improvements
Deliver Competitive Advantage
Access to Cutting Edge Research in Collaboration with Top Universities
Reliance
Available for all these languages:

## Fortran | C | C++ | Python | Java | MATLAB | C# and .NET | Excel and VBA | SMP & Multicore | Xeon Phi | Other languages & environments

NAG optimization solvers application example

The Fixed Income and FX team at Första AP-fonden use solvers from the NAG Library for optimizations and interpolations. Their system is highly dependent on the speed and accuracy of the NAG Library. For example, advanced yield curve modelling within the system uses a number of NAG functions; these yield curves are essential for the management of the Fixed Income portfolio. In addition to this application, the NAG Library is used for optimizing fixed duration indices; another tool that the management of the Fixed Income portfolio relies on.

“As long time users of NAG, both as developers and as end users, we are very satisfied with the performance of the Library and we are happy to recommend NAG to others.” Katarina Paschal, Portfolio Manager Fixed Income and FX at Första AP-fonden

# Services and Support

Outstanding service and support from our expert team
First line technical support when you need it

NAG’s Technical Support Service is provided by a team of specialists in numerical and statistical software development, in fact the NAG Library and Compiler development team share responsibility for the support of our software. We strongly believe that in order to effectively support complex software the technicians must be both experienced and understand the intricacies of the computational techniques. This conviction is reflected in the composition of the team most of whom are qualified to PhD level and have combined experience of software support in excess of 50 years.

NAG accuracy is quality assured

NAG provides accurate, documented, numerical software and services to help you make sure that your results are accurate. The validity of each NAG routine is tested for each platform that it is enabled for. Only when an implementation satisfies our stringent accuracy standards is it released. As a result, you can rely on the proven accuracy and reliability of NAG to give you the right answers. NAG is an ISO 9001 certified organization.

Reap the benefits of NAG expertise

The numerical codes that underpin the results from your software are not future proof. While the mathematics does not change, the codes have a limited lifespan because of new hardware structures, mathematical innovation and changes in application needs. NAG Numerical Services help you and your organization find and implement the optimum numerical computation solutions. NAG works with your team to impart skills and techniques that will help solve your numerical software problems.

Expert teaching and instruction from numerical experts

Your users, developers and managers can all benefit from NAG's highly regarded training courses. All of the training courses shown below have been delivered successfully either from NAG offices or at client premises. Training courses can be tailored to suit your particular requirements and be targeted to novice, intermediate or experienced levels. Specialized mentoring and development programs are also available for HPC managers.

Working in collaboration with industry and academia

NAG was founded on collaboration as an inter-University collaborative venture combining the talents of mathematicians and computer scientists. NAG has continued to collaborate with individuals and organizations over the past four decades and today longstanding and new partners are delivering tangible benefits to users and students all over the world.